Optimal. Leaf size=102 \[ \frac {8 b^2 x^{5 (m+1)}}{15 a^3 (m+1) \left (a+b x^{2 (m+1)}\right )^{5/2}}+\frac {4 b x^{3 (m+1)}}{3 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )^{5/2}}+\frac {x^{m+1}}{a (m+1) \left (a+b x^{2 (m+1)}\right )^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {271, 264} \[ \frac {8 b^2 x^{5 (m+1)}}{15 a^3 (m+1) \left (a+b x^{2 (m+1)}\right )^{5/2}}+\frac {4 b x^{3 (m+1)}}{3 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )^{5/2}}+\frac {x^{m+1}}{a (m+1) \left (a+b x^{2 (m+1)}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {x^m}{\left (a+b x^{2+2 m}\right )^{7/2}} \, dx &=\frac {x^{1+m}}{a (1+m) \left (a+b x^{2 (1+m)}\right )^{5/2}}+\frac {(4 b) \int \frac {x^{2+3 m}}{\left (a+b x^{2+2 m}\right )^{7/2}} \, dx}{a}\\ &=\frac {x^{1+m}}{a (1+m) \left (a+b x^{2 (1+m)}\right )^{5/2}}+\frac {4 b x^{3 (1+m)}}{3 a^2 (1+m) \left (a+b x^{2 (1+m)}\right )^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {x^{4+5 m}}{\left (a+b x^{2+2 m}\right )^{7/2}} \, dx}{3 a^2}\\ &=\frac {x^{1+m}}{a (1+m) \left (a+b x^{2 (1+m)}\right )^{5/2}}+\frac {4 b x^{3 (1+m)}}{3 a^2 (1+m) \left (a+b x^{2 (1+m)}\right )^{5/2}}+\frac {8 b^2 x^{5 (1+m)}}{15 a^3 (1+m) \left (a+b x^{2 (1+m)}\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 61, normalized size = 0.60 \[ \frac {x^{m+1} \left (15 a^2+20 a b x^{2 m+2}+8 b^2 x^{4 m+4}\right )}{15 a^3 (m+1) \left (a+b x^{2 m+2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 135, normalized size = 1.32 \[ \frac {{\left (8 \, b^{2} x^{5} x^{5 \, m} + 20 \, a b x^{3} x^{3 \, m} + 15 \, a^{2} x x^{m}\right )} \sqrt {b x^{2} x^{2 \, m} + a}}{15 \, {\left ({\left (a^{3} b^{3} m + a^{3} b^{3}\right )} x^{6} x^{6 \, m} + a^{6} m + a^{6} + 3 \, {\left (a^{4} b^{2} m + a^{4} b^{2}\right )} x^{4} x^{4 \, m} + 3 \, {\left (a^{5} b m + a^{5} b\right )} x^{2} x^{2 \, m}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b \,x^{2 m +2}+a \right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 54, normalized size = 0.53 \[ \frac {x^{m+1}\,\left (\frac {8\,b^2\,x^{4\,m+4}}{15}+a^2+\frac {4\,a\,b\,x^{2\,m+2}}{3}\right )}{a^3\,{\left (a+b\,x^{2\,m+2}\right )}^{5/2}\,\left (m+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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